Monday, 26 October 2009

Towards a Teleological Logic (Part 2)

How are we to formerly represent the claim that the purpose of the human eye is to perceive visual stimuli? One suggestion, which will ultimately prove insufficient, may be put as follows: Let E refer to the set of human eyes, and let P refer to the set of things that perceive visual stimuli. The claim that the purpose of the human eye is to perceive visual stimuli may be formerly represented as follows:

(1.1) (∀x)(E(x) → P(x))

According to (1.1), for something to be a member of the set of human eyes is sufficient for that thing to be a member of the set of things that perceive visual stimuli. However, (1.1) clearly fails to capture what we mean when say that the purpose of the human eye is to perceive visual stimuli. Cases of blindness represent a counterexample to (1.1), but they do not represent a counterexample to the claim that the purpose of the eyes is to perceive visual stimuli. Thus, the former is not equivalent to the latter. The take home message seems to be that the claim that something has a telos allows for exceptional cases, and therefore cannot be represented by the universal quantifier. Another suggestion, which will also prove to be insufficient, is to replace the universal with an existential quantifier. This yields:

(1.2) (∃x)(E(x) & P(x))

(1.2) offers a clear advantage over (1.1) since it does not require that all members of the set of eyes also be members of the set of things that perceive visual stimuli. However, (1.2) also fails to capture what we mean when we say that the purpose of the human eye is to perceive visual stimuli since we can imagine a situation in which the former is false and the latter is true. For example, suppose that a global pandemic, a virulent eye-infection let us say, rendered everyone on earth blind. In such a case (1.2) would be false, and yet we would still wish to say that the telos of the human eye is to perceive visual stimuli.

I believe that (1.1) and (1.2) both fail because they attempt to represent the claim that the human eye has a certain telos by focusing solely on how things are in the actual world. However, I believe that our concept of what it means for something to have a purpose is an essentially modal notion; one that appeals to how things are in worlds other than the actual world.

In attempting to formerly represent our notion of purposiveness I will be taking as my starting point the accessibility relation introduced by Saul Kripke. Within the Libnizian framework, to say that φ is necessarily true means that φ obtains in all metaphysically possible worlds. By contrast, Kripke-style possible world semantics relativises the notion of necessary truth to a subset of the metaphysically possible worlds; namely, the set of accessible worlds. The upshot is that modal statements (it is necessary that φ, it is possible that φ) need not take the same truth value in all possible worlds.

For example, suppose that Δ is the only world accessible from Γ and that Γ and Δ are both accessible from Δ. Moreover, let us suppose that Δ ⊩ φ and that Γ⊮ φ. On the present model, it is necessarily true that φ relative to Γ since φ obtains in all worlds accessible from Γ. However, it is not necessarily true that φ relative to Δ since φ does not obtain in all worlds accessible from Δ. Significantly, Kripke-style semantics allows for the possibility that a given world may fail to be accessible from itself (as is the case with Γ but not the case with Δ in our preceding example). As we shall soon see, this feature of Kripke-style semantics will be crucially important when we attempt to formerly represent the concept of purposiveness. As has become standard, I will be defining the relation of accessibility as an (uninterpreted) binary relation R(Γ,Δ) that holds between possible worlds Γ and Δ just in case Δ is accessible from Γ. If we let Γ denote the actual world, then we have the following two fundamental translational schema for possible world sematics:

(2.1) □φ =def φ is true at every world Δ such that R(Γ,Δ)

(2.2) ◊φ =def φ is true at some world Δ such that R(Γ,Δ)

There are numerous applications of Kripke-style semantics. For example, in physics the accessibility relation is construed in terms of nomological accessibility. φ is nomologically necessary just in case φ is true at all possible worlds that are nomologically accessible from the actual world. In short, φ is true at all possible worlds that obey the physical laws of the actual world. In deontic logic, the accessibility relation is construed in terms of morally perfect worlds. φ is obligatory just in case φ obtains in all morally perfect worlds and permissible just in case it obtains in some morally perfect world.

An important difference between nomological necessity and obligatoriness (or deontic necessity) is that the class of nomologically accessible worlds includes the actual world (since the actual world is a member of the class of worlds that obeys the physical laws of the actual world), but the class of morally perfect worlds does not include the actual world (since the actual world is not a member of the class of morally perfect worlds). Thus, if we were to restrict the universe to the class of morally perfect worlds, the actual world would be omitted. The accessibility relation enables us to avoid this unwelcome result by allowing for imperfect moral worlds in our universe (a class that includes the actual world), while restricting deontic access to those worlds that are morally perfect.

The notion of purposiveness seems to fall somewhere between nomological necessity and obligatoriness. When applied to purposiveness, the accessibility relation may be seen as restricting access to the set of teleologically ideal worlds, defined as the set of worlds in which all aims are achieved, all functions are fulfilled and all purposes are realised. (Henceforth, I will refer to teleologically ideal worlds as T-worlds.) This yields the following fundamental translational schema for purposiveness:

(2.4) □φ =def φ is true at all T-worlds

(2.5) ◊φ =def φ is true at some T-world

Like nomological necessity, and unlike obligatoriness, purposiveness is a descriptive concept, it tells us something about the way the world actually is, and not merely about how the world ought to be. We may identify the descriptive dimension of purposiveness with the fact that an object’s purpose is determined by facts about the actual world. For example, in the case of a biological system, its purpose is determined by what that system was selected for in the actual world. In the case of a human artefact, its purpose is determined by the intentions of the human designer in the actual world. Thus, just as we can only tell which worlds are nomologically accessible by inquiring about which physical laws obtain the actual world, we can only tell which possible worlds are teleologically accessible by inquiring into what a biological system was selected for, or what an artefact was designed for in the actual world.

However, since purposes often go unfulfilled in the actual world, the actual world is not a member of the class of T-worlds. Consequently, there is also a prescriptive dimension to the concept of purposiveness. In this respect, purposiveness is like obligatoriness; both concepts construe the accessibility relation in terms of a set of worlds that excludes the actual world.

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The Reasoner

Today my self-esteem hit an all time low when I stumbled across an argument proving conclusively that I don't exist. I’m quite distressed. This probably explains why my jeans are so loose-fitting. What’s equally upsetting is that I learned that non-existence does not exempt one from local or federal taxes.